Definition

In particular if 𝒴\mathcal{Y} is the terminal object in Topos, hence the canonical base toposSet, we say that a topos 𝒳\mathcal{X} is a Hausdorff topos if 𝒳→𝒳×𝒳\mathcal{X} \to \mathcal{X} \times \mathcal{X} is a proper geometric morphism.

More generally, since there is a hierarchy of notions of proper geometric morphism, there is accordingly a hierarchy of separatedness conditions.